1. Introduction - Econometrics at Illinois - University of Illinois at ...

Roger Koenker and Zhijie Xiao 19and therefore tests can be based on K n as before. Note that in this case estimationof _g provides as a byproduct an estimator of the function ' 0 (t) which is needed tocompute the process ^v n (t).In applications it will usually be desirable to restrict attention to a closed interval[ 0 ; 1 ] (0; 1). This is easily accommodated, following Koul and Stute (1999),Remark 2.3, by considering the modied test statistic,K n =sup jj~v() , ~v( 0 )jj= p 1 , 0 ;2[ 0 ; 1 ]which converges weakly, just as in the unrestricted case, to sup [0;1]jjw 0 ()jj. Thisrenormalization is useful in our empirical application, for example, since we are restrictedat the outset to estimating the ^v process on the subinterval [ 0 ; 1 ]=[:2;:8].Indeed, it may be fruitful to consider other forms of standardization as well.5.3. The location shift hypothesis. An important special case of the locationscaleshift model is, of course, the pure location shift model,F ,1y i jx i(jx i )=x > i + 0F ,10 ()This is just the classical homoscedastic linear regression model,y i = x > i + 0 u iwhere the fu i g are iid with distribution function F 0 . This model underlies much ofclassical econometric theory and practice. If it is found to be appropriate then it isobviously sensible to consider estimation by alternative methods. For F 0 Gaussian,least squares would of course be optimal. For F 0 unknown one might consider theHuber M-estimator, or its L-estimator counterpart,Z 1,^ =(1, 2) ,1 ^()d;see Koenker and Portnoy (1987). In the location shift model it is also well-knownfrom Bickel (1982), that the slope parameters, ( 2 ;:::; p ), are adaptively estimableprovided F 0 has nite Fisher information for the location parameter. Thus, it wouldbe reasonable to consider M-estimators like those described in Hsieh and Manski(1987) or the adaptive L-estimators described in Portnoy and Koenker (1989).The location-shift hypothesis can be expressed in standard form,R() =r;by setting R =[0.I p,1]; r =( 2 ;::: ; p ) > . It asserts simply that the quantile regressionslopes are constant, independent of. Again, the unknown parameters infR; rg are easily estimated so the process ^v n () is easily constructed. The transformationis obviously somewhat simpler in this case since g(t) =(t; ' 0 (t)) has one fewercoordinate than in the previous case.